Functions satisfying elementary relations

Author:
Michael F. Singer

Journal:
Trans. Amer. Math. Soc. **227** (1977), 185-206

MSC:
Primary 12H05

DOI:
https://doi.org/10.1090/S0002-9947-1977-0568865-2

MathSciNet review:
0568865

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Abstract: In this paper we deal with the following problems: When do the solutions of a collection of differential equations satisfy an elementary relation, that is, when is there an equation of the form where *R* is some algebraic combination of logarithmic, exponential and algebraic functions involving solutions of our differential equations? If such relations exist, what can they look like? These problems are given an algebraic setting and general forms for such relations are exhibited. With these, we are able to show that certain classes of functions satisfy no elementary relations.

**[1]**James Ax,*On Schanuel’s conjectures*, Ann. of Math. (2)**93**(1971), 252–268. MR**0277482**, https://doi.org/10.2307/1970774**[2]**Claude Chevalley,*Introduction to the Theory of Algebraic Functions of One Variable*, Mathematical Surveys, No. VI, American Mathematical Society, New York, N. Y., 1951. MR**0042164****[2a]**Robert C. Gunning and Hugo Rossi,*Analytic functions of several complex variables*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR**0180696****[3]**Irving Kaplansky,*An introduction to differential algebra*, Actualités Sci. Ind., No. 1251 = Publ. Inst. Math. Univ. Nancago, No. 5, Hermann, Paris, 1957. MR**0093654****[4]**Joseph Fels Ritt,*Integration in Finite Terms. Liouville’s Theory of Elementary Methods*, Columbia University Press, New York, N. Y., 1948. MR**0024949****[5]**J. F. Ritt,*On the integrals of elementary functions*, Trans. Amer. Math. Soc.**25**(1923), no. 2, 211–222. MR**1501240**, https://doi.org/10.1090/S0002-9947-1923-1501240-7**[6]**Maxwell Rosenlicht,*Liouville’s theorem on functions with elementary integrals*, Pacific J. Math.**24**(1968), 153–161. MR**0223346****[7]**Maxwell Rosenlicht,*On the explicit solvability of certain transcendental equations*, Inst. Hautes Études Sci. Publ. Math.**36**(1969), 15–22. MR**0258808****[8]**-,*On Liouville's theory of elementary functions*, Pacific J. Math. (to appear).**[9]**M. Singer,*Functions satisfying elementary relations*, Thesis, Univ. of California, Berkeley, 1974.

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DOI:
https://doi.org/10.1090/S0002-9947-1977-0568865-2

Article copyright:
© Copyright 1977
American Mathematical Society